1994, Vol. 6, No.1, 23-36
Mathematics Education Research Journal
Mathematics in the Lower Primary Years:
A Research-based Perspective on Curricula
and Teaching Practice
Bob Wright
Southern Cross University
Drawing on current research the author explicates twelve assertions relating to
curricula, teaching, learners and learning environments in lower primary school
mathematics. Topics discussed include: unchanging and under-challenging
curricula; the need for greater emphasis on developing children's verbal number
strategies and number sense, and on activities specifically suited to prenumerical
children; curriculum constraints on teachers; the' role of problem solving and
differing interpretations of problem solving; the need for a better understanding of
how children learn mathematics; differences in children's knowledge;
"anti-interventionism," discovery learning,constructivism, children's autonomy
and developmental learning; the need for compensatory programs; and learning in
collaborative settings. The author concludes that learning and teaching lower
primary mathematics continues to be an important area of focus and challenge for
teachers and researchers.
Children's difficulties with and dislike for mathematics seem to increase as
they progress to higher grades and, as testing becomes more significant in terms of
its immediate and longer term consequences. In Australia, where there is little
formal testing in the lower primary years, children's failures' become an
increasingly important issue from around Year Three onwards. Given these
circumstances it is tempting to conclude that all is well with mathematics in the
lower primary years, but it is important to distinguish between the symptoms and
causes of problems in higher grades.
There can be no doubt that the first three years of school (K-2) have a profound
effect on the rest of the child's mathematical education, because it is in the first
three years that the child first experiences success or failure, interest or boredom,
challenge or frustration. Further, at the end of the first three years of school, vast
differences are apparent in children's ability in, enthusiasm for, and beliefs about
mathematics. Thus mathematics in the lower primary years continues to be an
important area for research endeavours. In this article I have described aspects of
mathematics in the lower primary years which I regard as problematic. My
purpose in so doing is to indicate some areas where I believe future research efforts
could be profitably directed, and some issues that are important for any future
curriculum reviews, or initiatives in teacher or school development.
Some of the points discussed in this article apply specifically to the lower
primary years. Other points may apply to higher levels as well as the lower
primary years, although not necessarily in exactly the same way. Educational
significance rather than specificity determined inclusion.
Wright
24
Content of Curricula in the Lower Primary Years
1. Curriculum emphases in early number and related topics have not
undergone major changes in the last 30 years and have not changed
significantly in the last 10 years.
.'
Several Australian states produced new mathematics curricula in the late 1980s
(e.g., New South Wales Department of Education (1989); Queensland Department
of Education (1989); Victorian Ministry of Education (1988); Western Australia
Department of Education (1989); and more recently, A National Statement on
Mathematics for Australian Scools has been produced (Australian Education Council,
1991). With respect to early number and related topics there are no more than
minor differences among these documents. Additionally, there appears to be a
general consensus among teachers that, although new curricula embody many
changes in approaches and content in the middle and upper primary years there
have been no major changes in either approach or content in the lower primary
years. The most recent major change in the teaching of early number involved a
de-emphasis on the use of Cuisenaire rods and, associated with this, are-emphasis
on counting (Wright, 1992c) which occurred 10-15 years ago. That change aside,
much of the content and teaching approaches for early number and related topics
has remained unchanged for around 30 years and has its origins in the Cuisenaire
movement and Piagetian theory (Wright, 1992c).
2. In the first year of school many children are under-challenged in
mathematics/ particularly in the number area.
There is compelling evidence for this proposition on at least two fronts. Firstly,
research results reported by Young-Loveridge (1989) and Wright (1991c, 1994a)
strongly indicate this. Wright (1991c) assessed the arithmetical knowledge of 15
children from each of two beginning Kindergarten classes and argued:
The observation that almost all of the children from the higher S.B. [socio-economic]
kindergarten class were facile with the number word sequence in the range one to ten
and were beyond the stage of perceptual counting suggest1? that the prenumber and
early number topics typically undertaken in the first six months of kindergarten, as
indicated by state curricula and text books, are inappropriate for such children. (p. 9)
Young-Loveridge/s (1989) study in New Zealand involved interview-based
assessment of 81 children who were beginning school and, one year later, asking
their teachers whether they had taught particular concepts during the preceding
year. She observed that:
The fact that large numbers of children were taught certain concepts (e.g. rote
counting, enumeration, pattern recognition, ordinal numbers, numeral recognition)
even though they already knew them, but were not taught addition and subtraction
which they could also do, is a finding which is consistent with the idea that the
curriculum is not well matched to the skills of children. (p. 60)
Further, interviews conducted in 1991 by the author (Wright, 1992a), with 40
It
s
.t
a
J1
l,
:l.
:l
if
Mathematics in the Lower Primary Years: A Research-based Perspective
25
infants teachers teaching in the first year of. school indicate that many teachers,
particularly those with extensive teaching experience, are well aware of this. The
stat~of affairs just described can be summarised by saying that many children are
under-challenged. The point at issue here relates closely to learning through
problem solving which is discussed later in this article. That problem solving can
and should playa central role in the advancement of young children's arithmetical
knowledge has been readily demonstrated (e.g. Steffe, von Glasersfeld, Richards &
Cobb, 1983; Steffe & Cobb, 1988; Cobb, Wood & Yackel, 1991; Yackel, Cobb, Wood,
Wheatley & Merkel, 1990; Yackel, Cobb & Wood, 1991). Engaging a child in
arithmetical problem solving necessarily involves presenting a task which is a
genuine problem for the child, that is, a task for which the child does not have a
readily available procedure. The work of Steffe and colleagues is replete with
examples of problem solving of this kind (see also Wright, 1991a; 1991b).
e
3. Developing students' number sense and mental computation-
if
currently emphasised in the middle and upper primary years - needs to be
addressed in the lower primary years as well.
:l
5
.,
.
5
2
,
,
,)
In recent years, national reports and new curricula (e.g. Australian Education
Council, 1991, p. 109; National Council of Teachers of Mathematics' Commission
on Standards for School Mathematics, 1989, pp. 38-40; National Research Council,
1989, p. 82; New South Wales Department of Education, 1989, p. 13) have called for
an increased emphasis on developing students' number sense (Sowder, 1992) and
mental computation skills. Typically, this call is accompanied by recommendations
for a reduced emphasis on traditional written computation. This can be regarded as
a significant change of focus in the teaching of arithmetic in the middle and. upper
primary years, and has important implications for the teaching of arithmetic in the
lower primary years which, I believe, have not been adequately addressed.
In the lower primary years in New South Wales schools, the typical approach is
to introduce formal addition in the second year and to move almost immediately to
written forms, either as horizontal number sentences or in a so-called vertical
format. It is as if the idea of children adding and subtracting is synonymous with
children writing formal addition and subtraction statements, that is, with numerals
and signs for addition, subtraction and equals. In due course this is extended to
include written representations of the addition and subtraction of two-digit
numbers. What is' omitted in this sequence is a period during which children
develop and use verbal strategies to add and subtract two-digit numbers. The
underlying assumption seems to be that verbal or mental arithmetic is important
only to the extent to which it acts as a transition between arithmetic using materials
and written arithmetic (see also Wright, 1992c, pp. 128-9).
A key idea underlying the typical approach is that, in the range one to 100,
concepts of tens and ones, that is place value concepts, should be taught prior to
and separate from addition and subtraction. This approach is very clearly
embodied in the specification of numeration as a topic separate from and prior to
addition, subtraction and so on, which appears in the curriculum and prominent
texts series. An alternative approach (Steffe & Cobb, 1988), involves children
developing increasingly sophisticated verbal strategies for adding and subtracting.
26
Wright
An intrinsic component of this approach is the development of increasingly
sophisticated concepts of ten, because the strategies children develop for adding
and subtracting in this context typically involve incrementing or decrementing by
tens and ones.
In a similar vein Kamii (1989) recommends "that children invent their own
procedures for two-digit addition and learn place value in the process" (p. 66). The
assertion that place value should be understood before the introduction of addition
and subtraction involving two-digit numbers~ may be justifiable in terms of a
logical analysis. However, such a practice would be difficult to justify from a
conceptual perspective-at least not one informed by an understanding of the
conceptual paths along which children's arithmetical knowledge is likely to
develop. Finally, in phylogenetic terms, it seems likely that verbal addition and
subtraction involving numbers beyond ten pre-dated place value. As Steffe and
Cobb (1988) have stated:
We must remember that'the sequence of number words in a given culture was built
up long before a mature system of numerals became established.... In school
mathematics there is an unresolved tension between children's verbal number
sequences and written numerals. Whatever numerical operations are available to
children are initially symbolized by their verbal nUn:J-ber sequences and these
operations do not serve as a basis for the written system.... The work with the
standard paper and pencil algorithms should be abandoned and replaced with work
on the schemes counting-on, counting-up-to, and counting-down-to. (p. 321)
The last sentence of this-excerpt would appear to have profound implications for
early childhood mathematics education.
4. There is a need to include in the Kindergarten year, number-related
activities which are likely to develop more sophisticated numerical thinking
in prenumerical children.
Prenumerical children are those who have not abstracted number (Steffe et al.,
1983, e.g., pp. 72-4; see also Wright, 1991a; 1991b) and are at either the perceptual
or figurative stage (Steffe & Cobb, 1988; Wright, 1991a). Wright (1991b, pp. 44-5)
provides detailed examples of activities appropriate for prenumerical children.
These . include ascribing number to spatial and auditory patterns, counting
sequences of sounds and movements, as well as a range of activities based on
forward and backward number word sequences. These and more advanced
activities specified for children who have constructed the initial number sequence
(pp. 45-6) should be considered as alternatives to the traditional prenumber and
early number topics which still appear in current curricula (e.g., Numeration 1-5 in
the New South Wales Department of Education's (1989) Mathematics K-6) because
the latter seem to assume that children cannot do simple sorting exercises or count
small collections.
Similarly, work on patterns specified at this level is apparently justified by the
rather vague argument that pattern is important in mathematics. Although most
would agree that pattern is commonly found in mathematics, does it therefore
follow that the most appropriate way to advance the mathematical understanding
of 5-year-olds is to have them copy patterns of colour and shape etc.?
Mathematics in the Lower Primary Years: A Research-based Perspective
27
Teacher Perspectives
5. Teachers are constrained by curriculum and inflexible expectations of
what is to be taught at each grade level:
On the basis of interviews with 40 teachers across the three eastern states it
appears that, in many schools, there exist implicit but very clear rules about which
topics should and should not be taught in a given year. Thus, in the first year of
school, number work extends to concepts of numbers up to about ten but this does
not include formal introduction of addition and subtraction. In the New South
Wales mathematics curriculum, this appears as Numeration Topics 1'-5 (e.g., New
South Wales Department of Education, 1989, p. 194), and is equivalent to topics
which, at an earlier time, were referred to as prenumber and early number. The
former includes activities such as sorting, matching, classifying and making
patterns, and the latter includes what used to be called cardinal and ordinal
number. Thus the typical number program for a Kindergarten class consists of the
topics just described and perhaps one or two of the early addition topics (e.g.
p. 210). When one considers that, at the beginning of their first year, many children
can use counting in relatively sophisticated ways (Wright, 1991c; 1994a), to solve
additive and subtractive tasks, this seems a rather severe limitation.
6. Teachers need exemplars ofwhat it means to involve 5- to 7-year-olds in
mathematical problem solving.
Within Australia in recent years, many writers have underlined the need to
engage young children in mathematical problem solving (e.g., English, 1990;
Groves & Stacey, 1990). Additionally, the importance of mathematical problem
solving is emphasised in new curricula (e.g. New South Wales Depertment of
Education, 1989, pp. 22-4) and in the nationally developed documertt A National
Statement on Mathematics for Australian Schools (Australian Education Council,
1991). Indeed, it seems reasonable to claim that a majority of classroom teachers
are well aware of this emphasis on problem solving. Nevertheless, many of these
teachers do not have access to exemplars of teaching involving mathematical
problem solving. Further, in the first year of school there are barriers to adopting
teaching methods typically associated with an emphasis on problem solving, for
example, collaborative group work followed by whole class discussion of a range
of solutions (e.g. Yackel et aI., 1990; 1991). Obvious barriers are that the children
cannot read instructions or write solutions, and may not have been well socialised
into group activities.
7. Problem solving needs to be seen as the means oflearning arithmetic as
well as an end result of the learning ofarithmetic.
Many teachers tend to see problem solving as an end rather than a means of
mathematics teaching (Wright, 1992a). Closely related to this is the tendency to
cast the learning of arithmetic as preparatory to and somewhat sep3.rate from
problem solving in school mathematics (Groves & Stacey, 1990, p. 10-11). I agree
28
Wright
with Groves and Stacey that "children need to ... investigate appropriate
mathematical situations and tackle non-routine problems in order to develop
strategies and skills for dealing with such situations in real life" (p. 10).
Nevertheless arithmetic which, from an adult's perspective, "relates to specific
ite,ms of knowledge or skills" (p. 10) can and should be taught through problem
solving and should not be cast as traditional mathematics and complementary to
open-ended, investigative activities (p. 11). By way of contrast, problem solving
should be regarded as an integral part of the learning of arithmetic. It is through
the activity of problem solving that children develop more sophisticated
arithmetical meanings and strategies, and this is the basis of children's
advancements in arithmetic.
The essential feature of this problem solving activity is' that children are
challenged to change their current strategies, and to re-organise their arithmetical
thinking. Of course these re-organisations do not necessarily happen quickly and
thus patience is required. Current mathematics curricula (e.g., New South Wa,es
Department of Education, 1989, p. 23) emphasise teaching for, about and through
problem solving. Teaching through problem solving, however, needs to be further
developed and better understood. One aspect of this is over-emphasis on the role
of so-called "real-world problems." Two assumptions which seem to be implicit in
dominant text series, and which I would challenge are: (a) it is necessary for
children to solve real-world problems to understand arithmetic properly; and (b)
engaging children in problem solving necessarily involves children in solving
real-world problems. These issues are closely related to what I would describe as
current over-emphases on utility and societal needs as reasons for teaching
mathematics in schools (Galbraith, 1988).
A major source, I believe, of the emphasis in the last 15 years or so on the use of
real-world problems was research in the late 1970s which showed that, in applied
settings such as shopping, 14-year-olds for example, used very simple arithmetical
strategies such as counting and addition rather than more advanced arithmetical
methods (e.g., Hart, 1983, p. 120). From findings of this kind, I believe curriculum
developers formed the view that classroom mathematics should focus to a much
greater extent on mathematics in real-life situations. An alternative view is that the
reason many 14-year-olds did not use more advanced mathematics in real-life
settings was because they lacked understanding of the more advanced
mathematics. From this view it follows that the key to the problem is to find ways
of ensuring that students understand mathematics better. My work in several
different longitudinal studies focusing on early mathematics learning, confirms the
view that the key to developing understanding is to engage children in solving
challenging mathematical problems which involve thinking hard and reflecting on
their mathematical activity. I do not object to topics involving the application of
mathematics to real life situations being included in mathematics curricula. The
view that I wish to challenge is that such applications are essential for
understanding and that mathematics curricula should consist predominantly of
such applications..
An explication of an approach entitled 'realistic mathematics education'
(Streefland, 1991) provides interesting perspectives on some of the issues discussed
in this section. For example, relevant to the discussion above of problem solving as
It
Mathematics in the Lower Primary Years: A Research-based Perspective
e
a means to learning is Treffers' (1991, e.g. pp. 26-56) discussion of the empiristic,
mechanistic, structuralistic and realistic directions in mathematics education. _.
:>
29
I.
C
1.
::>
:Y
~
1.
i
I
L
8. Teachers need and are seeking a better understanding ofhow young
children learn mathematics.
Naturally enough, teachers are very experienced and able at planning
mathematical activities and organising children to participate in activities.
Nevertheless, teachers seem to be much less able to judge how much and what
·kind of learning is taking place during these activities. Indeed teachers express
concern that, either they cannot be sure whether useful learning is occurring or that
they suspect that not very much learning is occurring but are not sure about what
changes to make. Thus it would seem appropriate to develop further our
understanding of the learning of mathematics, as it occurs in classroom settings,
and make this knowledge accessible to teachers.
l
Individual Differences and Learning Environments
in the Lower Primary Years
9. Insufficient account is taken of the qualitative differences in children's
number knowledge.
Recent research (Wright, 1991c; 1994a) has revealed a very wide range in the
number knowledge of children beginning the first year of school. This three-year '
difference (1991c, pp. 11-12) means, for example, that a given Kindergarten class
includes children who use relatively sophisticated strategies such as counting-on to
solve additive and subtractive tasks, are facile with number words beyond one
hundred," and can identify two and three digit numerals, as well as children who
cannot enumerate small collections and cannot say the number words to "ten."
That such differences are common has been demonstrated in the study by
Young-Loveridge (1991, e.g., pp. 12-17). There would appear to be no simple
answers to the question of how best to take account of these differences. One
might ask: are there available to teachers, strategies which take account of and
intentionally cater for these differences?
If
10. Other assumptions underlying lower primary school mathematics.
A range of interrelated and important notions warrant further consideration by
mathematics educators with interests in the lower primary years. These notions
include readiness (Young-Loveridge, 1987), developmental learning (Clay, 1991b),
children's autonomy and 'anti-interventionism' (Young-Loveridge, 1988),
constructivism and discovery learning (Cobb, 1991). Young-Loveridge (1988) has
criticised the view that, in the interests of developing children's autonomy, teachers
should intervene as little as possible in children's mathematics learning, and warns
that "the likely consequence of sitting back and waiting for children to discover
mathematics on their own is that lower levels of mathematics achievement will
30
Wright
eventually be attained" (p. 39). Cobb (1991) has criticised naive interpretations of
constructivism and discovery learning. For example, he criticised the views that (a)
mathematics concepts can be codified in materials or pictures for discovery by
children (pp. 6---7); and (b) because children must necessarily mentally construct
mathematics for themselves teachers should leave children virtually on their own to
"discover" through activity-based learning (p. 8).
Clay (1991b) discussed a range of issues about early literacy learning and many
of these seem equally relevant to mathematics. For example, how useful is an
assumption that children's learning follows a prescribed developmental path? How
should experts (such as teachers) interact with learners (pp. 263-5)? Is a child's
passive approach to learning developmental or learned and do some teachers use a
notion of developmental learning to justify not actively assisting passive children to
learn (p. 271)? Finally, how useful is a grow-first learn-later hypothesis about
development (p. 272)?
Some of the notions just described are linked closely to trends in the teaching of
early childhood mathematics which also, I believe, warrant examination from a
research perspective. In turn, these trends relate closely to the discussion on the
development of number sense, and on learning through problem solving presented
earlier in this paper. These trends include the views that: (a) early arithmetical
learning is less important than it was considered to be in the past and thus deserving
of less class time (e.g., space, number and measurement should be allocated equal
amounts of class time); (b) early mathematics can and should be learned wholly or
mainly through thematic approaches, games or in learning situations that arise
opportunistically; (c) manipulatives, activity and group discussion are the
sufficient and key ingredients of good environments for early mathematics learning
(see Gravemeijer (1991) for an interesting review of the use of manipulatives);
(d) habituation of simple arithmetic combinations (i.e., basic facts) and
automatising of procedures (when judged to be appropriate from the perspective of
the learner's knowledge) are unimportant or counterproductive; (e) it is
counterproductive to view mathematical knowledge and in particular, arithmetical
knowledge as essentially hierarchical.
The trends just listed do not arise spontaneously in the minds of classroom
teachers. Yet all of them are evident in classroom practice, possibly because they are
implicitly or explicitly conveyed to teachers through textbook guides, curricula,
teacher education courses, and so on.
11. Current teaching approaches do not readily reveal those children who
require a special intervention or compensatory program.
It is interesting to consider why there have not been more prominent calls for
early intervention in mathematics. One possible explanation is that current
teaching approaches and curriculum constraints work against teacher awareness of
the differences in children's number knowledge. For example, in New South Wales
(e.g., New South Wales Department of Education, 1989) it seems that, in many
schools, the mathematics program in the infants school dictates that formal work
on addition and subtraction is not done in the first year. In the second year of
school initial number work focuses on addition in the range one to ten and, for the
Mathematics in the Lower Primary Years: A Research-based Perspective
31
vast majority of children at this levet this is not a difficult topic (Wright, 1991c;
1994a). Mandatory basic skills testing occurs in NSW government schools in Year
Three-the year following the three years of infants' school (e.g. Doig & Masters,
1992). One wonders about the apparent hiatus from lower to middle primary years
in the assessment of children's progress. Thus in the infant years notions such as
'developmentally appropriate learning' and the importance of "readiness" are
used to explain away the 'three-year-difference' (Wright, 1991c) between lowest
and highest achievers.
In Year Three, on the basis of scores in the basic skills test, the same lowest
achievers are labelled as Band I-scoring in the lowest quartile and requiring
remediation, but more than likely destined for chronic failure and membership of
the "couldn't do maths at school" club. I am currently directing an applied
research and development project aimed at intervention in early arithmetical
learning (Wright, 1992b; 1993a, b; 1994b; Wright, Stanger, Cowper & Dyson, 1993;
Wright, Cowper, Stafford, Stanger, & Stewart, 1994). This project has involved the
development of a mathematics recovery program which, in organisational terms, is
similar to the Reading Recovery program (e.g., Clay, 1987; 1991a). Some key
features of the mathematics recovery program are: (a) intensive, individualised
teaching of low-attaining first-graders by specialist teachers for teaching cycles of
up to 20 weeks; (b) an extensive professional development course to prepare
specialist teachers, and on-going collegial and leader support for these teachers; (c)
use and further development of a strong underpinning theory of young children's
mathematical learning (e.g., Steffe et aI., 1983; Steffe & Cobb, 1988); and (d) use of
especially developed instructional activities and assessment procedures.
The project has, I believe, achieved significant results in terms of children's
advancements and teachers' professional learning (see Wright, 1993a; Wright et aI.,
1994). Young-Loveridge (1993) and Peters (1993) have researched approaches to
early mathematics intervention which are somewhat different from the approach
developed by Wright and colleagues. The approaches studied by Young-Loveridge
did not feature individualised teaching or extensive teacher development. Rather
they involved using number books and games and were classroom-based, or
involved withdrawing pairs or working with parents. Of relevance to a
consideration of the effectiveness of these intervention programs are Slavin and
Madden's (1989) three principles of effective programs for students at risk: (a) the
programs are comprehensive, that is "complete, systematic, carefully constructed
alternatives to traditional methods" (p. 11); (b) the programs are intensive, that is
they are individualised and use one-to-one tutoring rather than group instruction;
and (c) the programs "frequently assess student progress and adapt instruction to
individual needs" (p. 11).
12. Then: is a need to understand how mathematics arises in settings
involving collaborative learning.
I have recently completed the field aspects of a longitudinat classroom-based
research project in the Kindergarten year (Wright, 1993b). This study applied a
research methodology used by Cobb and colleagues at second and third grade
levels (e.g. Cobb et aI., 1991; Yackel et aI., 1990; Yackel, et aI., 1991) and involved a
32
Wright
research team working collaboratively with a classroom teacher to plan and
evaluate three lessons per week for approximately half of the school year. One
focus of this study which will be the subject of a future report is 5-. and
6-years-olds' collaborative solutions of arithmetical problems while working in
pairs. Of interest are the strategies that children generate, whether such strategies
arise communally or individually, and children's processes of negotiation in these
settings. The study shows that in the first year of school, children working in pairs
can learn arithmetic collaboratively, and has provided insight into the kinds of
strategies that are likely to arise in collaborative settings. Collaborative problem
solving in pairs was one of three instructional approaches used in the project. The
others were whole class or large group instruction and teacher-orchestrated
discussion of solutions to problems which children had previously attempted to
solve either individually or in pairs. As a result of the project we have concluded
that all three instructional approaches have an important role in early mathematics
programs.
Problem solving activity should not be exclusively collaborative-there is an
important place for individual problem solving as well. Of relevance to a
consideration of effective settings for collaborative learning are two recent studies
in New Zealand both of which were Ministry funded (Higgins, 1993; 1994;
Thomas, 1993). In reporting her study Higgins points out that games are a
prominent feature of early mathematics programs in New Zealand, raises
questions about the effectiveness of some of these games and identifies some
necessary features of effective small group instruction. As pointed out by Higgins,
several writers have strongly advocated using games in early mathematics
programs (e.g. Kamii, 1985).
An important issue for researchers is that there may well be better alternatives
than the extensive or exclusive use of games. I use "better" in the sense that these
alternatives would be more effective in terms of the extent of children's
mathematics learning and more informed by the recent results of research. Thomas
(1993) documents changes associated with an early mathematics program, over the
course of an action research project of duration two terms. These changes related
to a teacher's classroom practice and the nature of children's talk during
teacher-independent group work. The action research program involved: (a)
viewing videotapes of the group work; and (b) fortnightly meetings of the
researcher, the teacher, and a second teacher participating in the study. After the
initial phase of the project and as a result of viewing the videotapes the teacher
made two major changes to her approach to the group work. These were to adopt
a more structured topic-based approach, and to have the children work in pairs
rather than in groups of around seven. Working in pairs rather than larger groups
was used almost exclusively in the studies by Cobb et al. (1991) as well as in
Wright's (1993b) Kindergarten year study.
In these studies it was typical for students to work on a common set of
problems in the same sequence. What seems apparent is that the teacher in
Thomas' study, as a result of reflection on practice in an action research project,
changed the group work practices in her classroom to practices closely attuned to
those used in the studies by Cobb et al. (1991) and by Wright (1993b). Thomas
argued that, as a result of these changes, children's cognitively-oriented talk during
(
Mathematics in the Lower Primary Years: A Research-based Perspective
'33
group work increased from 27% to 63%, although of this cognitively-oriented talk,
only 4% was "abstract talk"-involving the clarification of one's ideas-while the
remaining 59% was "action talk"related to the activity of the moment. Thomas'
study raises important issues about the relative usefulness of different approaches
to collaborative learning.
Conclusion
The list of issues discussed in this article is not intended to be exhaustive.
Underlying this article is the view that lthe earning and teaching of lower primary
mathematics continues to be an important area of focus and challenge for
researchers and teachers. The aim of this paper has been to take up this challenge
and to identify specific issues which need further research.
Acknowledgments
This article is an outcome of research projects funded by Australian Research
Council Large Grant No. A79131881, Australian Research Council Collaborative
Grant No. AM9180064, contributions in kind from government and Catholic school
systems of the North Coast Region of NSW, and additional research grants from
Southern Cross University. I am grateful to Margaret Cowper, Rosalie Dyson,
Garry Stanger, Rita Stewart and to the reviewers of this paper whose comments
and suggestions have enabled me to clarify the ideas expressed here.
References
Australian Education Council. (1991). A national statement on mathematics for Australian
schools. Melbourne: Curriculum Corporation.
Clay, M. (1987). Implementing Reading Recovery: Systemic adaptations to an educational
innovation. New Zealand Journal of Educational Studies, 22(1), 35-58.
Clay, M. (1991a). Becoming literate: the construction of inner control. Auckland, NZ:
Heinemann.
Clay, M. (1991b). Developmental learning puzzles me. AustralianJournal of Reading, 14(4),
262-275.
Cobb, P. (1991). Reconstructing elementary school mathematics. Focus on Learning Problems
in Mathematics, 13('2),3-32.
Cobb, P., Wood, T., & Yackel, Y. (1991). A constructivist approach to second grade
mathematics. In E. von Glasersfeld (Ed.) Radical constructivism in mathematics education.
Dordrecht, The Netherlands: Kluwer.
Doig, B. & Masters, G. (1992). Through children's eyes: A constructivist approach to
assessing mathematics learning. In G. Leder (Ed.) Assessment and learning of
mathematics (pp. 269-289). Melbourne: ACER.
English, E. (1990). Mathematical power in early childhood. Australian Journal of Early
Childhood, 15(1),37-42.
Galbraith, P. (1988). Mathematics education and the future: A long wave view of change.
For the Learning of Mathematics, 8(3), 27-33.
Gravermeijer, K. (1991). Didactical background of a mathematics program. for primary
education. In L. Streefland. (Ed.). Realistic mathematics in the primanJ school (pp. 57-76).
Utrecht, The Netherlands: Utrecht University.
Wright
34
Groves, S. & Stacey, K. (1990). Problem solving-a way to link mathematics to young
children's reality. Australian Journal of Early Childhood, 15(1),5-11.
Hart, K. (1983). I know what I believe; Do I believe what I know? Journal for Research in
.
Mathematics Education, 14, 119-125.
Higgins, J. (1993). "This isn't a winning game": The effectiveness of independent mathematics
activities in junior primary classrooms. Paper presented at the annual conference of the
New Zealand Association for Research in Education, Hamilton, 2-5 Dec.
Higgins, J. (1994). Learning to cooperate: Small group interaction in New Zealand elementary
mathematics classrooms. Paper to be presented at the Seventeenth Annual Conference of
the Mathematics Education Research Group in Australasia, Lismore, NSW, 5-8 July. [to
appear in the conference proceedings]
Kamii, C. (1985). Young children reinvent arithmetic. New York: Teachers College Press.
Kamii, C. (1989). Young children continue to reinvent arithmetic. New York: Teachers College
Press.
National Council of Teachers of Mathematics' Commission on Standards for School
Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston,
VA: Author.
National Research Council. (1989). Everybody counts: A report to the nation on the future of
mathematics education. Washington, DC: National Academy Press.
New South Wales Department of Education. (1989). Mathematics K-6. Sydney: Author.
Peters, S. (1993). Early mathematics intervention with seven-year-olds: The use of maths games in
the classroom. Paper presented at the annual conference of the New Zealand
Association for Research in Education, Hamilton, 2-5 Dec.
Slavin, R. E., & Madden, N. A. (1989). What Works for students at risk: A research synthesis.
Educational Leadership Feb., 4-13.
Sowder, J. T. (1992). Making sense of numbers in school mathematics. In G. Leinhardt, R.
Putnam, & R. A. Hattrup (Eds.), Analysis ofarithmetic for mathematics teaching. (pp. 1-51).
Hillsdale, NJ: Lawrence Erlbaum.
Steffe, L. P., von Glasersfeld, E., Richards, J. & Cobb, P. (1983). Children's counting types:
Philosophy, theory, and application. New York: Praeger.
Steffe, L. P., & Cobb, P. (1988). Construction of arithmetic meanings and strategies. New York:
Springer-Verlag.
Streefland. L., (Ed.). (1991). Realistic mathematics in the primary school. Utrecht, The
Netherlands: Utrecht University.
Thomas, G. (1993). Talking and learning: A case study ofjunior mathematics. Paper presented at
the annual conference of the New Zealand Association for Research in Education,
Hamilton, 2-5 Dec.
Treffers, A. (1991). Didactical background of a mathematics program for primary education.
In L. Streefland. (Ed.). Realistic mathematics in the primary school (pp. 21-56). Utrecht,
The Netherlands: Utrecht University.
Victorian Ministry of Education Schools Division. (1988) The mathematics framework P-I0.
1988. Melbourne: Author.
Western Australian Department of Education. (1989). Western Australian Mathematics
Curriculum. Perth: Author.
Wright, R. J. (1991a). An application of the epistemology of radical constructivism to the
study of learning. The Australian Educational Researcher, 18 (1), 75-95.
Wright, R. J. (1991b). The role of counting in children's numerical development. The
Australian Journal of Early Childhood, 16 (2),43-48.
Mathematics in the Lower Primary Years: A Research-based Perspective
1
5
I
f
)
35
Wright, R. J. (1991c). What number knowledge is possessed by children entering the
kindergarten year of school? The Mathematics Education Research Journal, 3 (1), 1-16.
Wright, R. J. (1992a). Concept development in early childhood mathematics: Teachers' theories and
research. Paper presented at the Seventh International Congress on Mathematical
Education, Quebec, Canada, 17-23 August.
Wright, R. J. (1992b). Intervention in young children's arithmetical learning: The development ofa
research-based Mathematics Recovery program. Paper presented at the joint conference of
the Australian and New Zealand Associations for Research in Education, Geelong,
22-26 November.
Wright, R. J. (1992c). Number topics in early childhood mathematics curricula: historical
background, dilemmas, and possible solutions. The Australian Journal of Education,
36(2), 125-142.
Wright, R. J. (1993a). An overview of a project organisationally similar to Reading Recovery
focusing on intervention in the mathematics learning ofat-risk six-year-olds. Paper presented
at the annual conference of the New Zealand Association for Research in Education,
Hamilton, 2-5 Dec.
Wright, R. J. (1993b). Problem-centred mathematics in the first year of school: Results from a
longitudinal,' classroom-based project. Paper presented at the annual conference of the
New Zealand Association for Research in Education, Hamilton, 2-5 Dec.
Wright, R. J. (1994a). A study of numerical development in the first year of school.
Educational Studies in Mathematics, 26, 25-44.
Wright, R. J. (1994b). Working with teachers to advance the arithmetical knowledge oflow-attaining
6- and 7-year-olds: First year results. Paper to be presented at the Eighteenth Annual
Conference of the International Group for the Psychology of Mathematics Education,
Lisboa, Portugal. [to appear in the conference proceedings]
Wright, R. J., Cowper, c., Stafford, A., Stanger, G., & Stewart, R. (1994). The mathematics
recovery project-A progress report: Specialist teachers working with low-attaining
first-graders. Paper presented at the Seventeenth Annual Conference of the
Mathematics Education Research Group in Australasia, Lismore, NSW, 5-8 July.
Wright, R. J., Stanger, G., Cowper, C. & Dyson, R. (1993). The Maths Recovery Project: An
initial report. Paper presented at the Sixteenth Annual Conference of the Mathematics
Education Research Group in Australasia, Brisbane, 9-13 July.
Yackel, E., Cobb, P & Wood, T. (1991). Small group interactions as a source of learning
opportunities in second grade mathematics. Journal for Research in Mathematics
Education, 22(5), 390-408.
Yackel, E., Cobb, P., Wood, T., Wheatley, G. & Merkel, G. (1990). The importance of social
interaction in children's construction of mathematical knowledge. In 1990 Yearbook of
National Council of Teachers of Mathematics (US). Reston, VA: NCTM.
Years 1-10 mathematics sourcebook. (1989). Brisbane: Queensland Department of Education.
Young-Loveridge, J. (1987). Learning mathematics. British Journal of Developmental
Psychology, 5 , 155-167.
Young-Loveridge, J. (1988). Is greater autonomy always in the best interests of children's
mathematics learning? Australian Journal of Early Childhood, 13(3), 37-40.
Young-Loveridge, J. (1989). The development of children's number concepts: the first year
of school. New Zealand Journal of Educational Studies, 24 (1), 47-64.
Young-Loveridge, J. M. (1991). The development of children's number concepts from ages five to
nine. Early mathematics learning project, Phase 2, Volumes 1 & 2. Hamilton, NZ:
University of Waikato.
36
Wright
Young-Loveridge, J. M. (1993). The effects ofearly mathematics intervention: The EM/-55 Study,
Volumes 1 & 2. Hamilton, NZ: University of Waikato.
..-..
_-.-.- - - - - - -
Author
Bob Wright, Centre for Education, Southern Cross University, P a Box 157,
Lismore, New South Wales 2480, Australia